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Theorem summodclem2 11403
Description: Lemma for summodc 11404. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
Hypotheses
Ref Expression
isummo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
isummo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
summodclem2.g 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
Assertion
Ref Expression
summodclem2 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2
Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5527 . . . . 5 (𝑚 = 𝑎 → (ℤ𝑚) = (ℤ𝑎))
21sseq2d 3197 . . . 4 (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑎)))
31raleqdv 2689 . . . 4 (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴))
4 seqeq1 10461 . . . . 5 (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
54breq1d 4025 . . . 4 (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
62, 3, 53anbi123d 1322 . . 3 (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
76cbvrexv 2716 . 2 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8 simplr3 1042 . . . . . . . . 9 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9 simplr1 1040 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ (ℤ𝑎))
10 uzssz 9560 . . . . . . . . . . . 12 (ℤ𝑎) ⊆ ℤ
119, 10sstrdi 3179 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ ℤ)
12 1zzd 9293 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 1 ∈ ℤ)
13 simprl 529 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℕ)
1413nnzd 9387 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℤ)
1512, 14fzfigd 10444 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ∈ Fin)
16 simprr 531 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
17 f1oeng 6770 . . . . . . . . . . . . . 14 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
1815, 16, 17syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ≈ 𝐴)
1918ensymd 6796 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ≈ (1...𝑚))
20 enfii 6887 . . . . . . . . . . . 12 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
2115, 19, 20syl2anc 411 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ∈ Fin)
22 zfz1iso 10834 . . . . . . . . . . 11 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
2311, 21, 22syl2anc 411 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24 isummo.1 . . . . . . . . . . . . 13 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
25 simplll 533 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26 isummo.2 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2725, 26sylan 283 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
28 eleq1w 2248 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
2928dcbid 839 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
30 simpr2 1005 . . . . . . . . . . . . . . 15 (((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴)
3130ad2antrr 488 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴)
32 simpr 110 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → 𝑘 ∈ (ℤ𝑎))
3329, 31, 32rspcdva 2858 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → DECID 𝑘𝐴)
34 summodclem2.g . . . . . . . . . . . . 13 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
35 eqid 2187 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑔𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑔𝑛) / 𝑘𝐵, 0))
36 simprll 537 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37 simpllr 534 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38 simplr1 1040 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑎))
39 simprlr 538 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
40 simprr 531 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4124, 27, 33, 34, 35, 36, 37, 38, 39, 40summodclem2a 11402 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
4241expr 375 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
4342exlimdv 1829 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
4423, 43mpd 13 . . . . . . . . 9 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45 climuni 11314 . . . . . . . . 9 ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
468, 44, 45syl2anc 411 . . . . . . . 8 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
4746anassrs 400 . . . . . . 7 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48 eqeq2 2197 . . . . . . 7 (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦𝑥 = (seq1( + , 𝐺)‘𝑚)))
4947, 48syl5ibrcom 157 . . . . . 6 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
5049expimpd 363 . . . . 5 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
5150exlimdv 1829 . . . 4 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
5251rexlimdva 2604 . . 3 (((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
5352r19.29an 2629 . 2 ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
547, 53sylan2b 287 1 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 835  w3a 979   = wceq 1363  wex 1502  wcel 2158  wral 2465  wrex 2466  csb 3069  wss 3141  ifcif 3546   class class class wbr 4015  cmpt 4076  1-1-ontowf1o 5227  cfv 5228   Isom wiso 5229  (class class class)co 5888  cen 6751  Fincfn 6753  cc 7822  0cc0 7824  1c1 7825   + caddc 7827   < clt 8005  cle 8006  cn 8932  cz 9266  cuz 9541  ...cfz 10021  seqcseq 10458  chash 10768  cli 11299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300
This theorem is referenced by:  summodc  11404
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